248. To prove that

loga (m") = (log.c m) x n. Law III.

logą (m/m) = (logą m) = n. Law IV.
Let logą m =a; this means oca = m.
Now m", i.e. (2)", = 24xn by Index Law III.
And m, i.e. */(2a), = 2a=n by Index Law IV.
Expressing these equations in Logarithmic language,

loga (m") =a x n, i.e. (log, m) x n.

logą (m/m) =a + n, i.e. (logam) = n. 249. The four laws just proved show that in each case the operation to be performed upon the indices or logarithms is arithmetically simpler than that to be performed on the powers.

Thus for multiplication of powers we substitute addition of logarithms;

for division of powers, subtraction of logarithms;
for power-raising of powers, multiplication of logarithms;
for root-taking of powers, division of logarithms.

250. In this simplification of arithmetical operations consists the value of Logarithms. Thus tables are published giving the logarithms of numbers to the base 10.

Suppose we had to find correctly to four decimal places the product 3.4764 x 7.6819. From the tables we should find that

log10 3.4764 = •5411297 and log10 7.6819 = •8854686. Now (by Law I.) log10 (3:4764 x 7.6819) = log10 3.4764 + logio 7:6819 = 1.4265983 by simple addition of the logarithms.

Again referring to the tables, we should find that 1.4265983 is the logarithm of 26.7053.

Hence 26-7053 is the required product.

Logarithms to different bases.

= a.

251. To show that logo a x loga b= 1.
Let logo a = x; this means be
Let loga b=y; this means ay = b.

From these two equations eliminate b, by raising the latter to the ath power.

Qay = b = a', .. wy=1,

i.e. logo a x loga b= 1.

= C.

252. To show that loga b logo c = loga C.
Let loga b = x; this means a* = b.
Let logo C= y; this means by

From these two equations eliminate b, by raising the former to the yth power. Thus

axy = by=C, .. xy = logac,

i.e. loga b logo c = loga c.

253. To transform from one to another system of logarithms.

logac By the last article, logo C =

loga b Hence, if we have given a system of logarithms to the base a, to find the logarithm of any number c to a new base b, we must divide the given logarithm of c in that system by the logarithm of the new base in that system.

Thus the logarithm of any number in the given system has to be divided by the same quantity logab, in order to find its logarithm to base b.


The constant multiplier

is called the Modulus of

loga 6

transformation from base a to base b.


254. In the decimal notation we express any number by means of the ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 which, when standing alone, stand for zero, one, two, three, &c., respectively. The decimal point' is placed after a digit, when it is necessary to indicate that it has its natural value. When digits are placed together, their value depends on their relative position. The 'units' place' being immediately before the decimal point', each removal of a digit to the left raises its value ten-fold; and hence each removal to the right lowers its value ten-fold. The number of removes from the units' place indicates the power of ten by which the digit has been multiplied. Digits to the left of the units' place correspond to the positive, those to the right correspond to the negative powers of ten. The units' place—not the decimal point—thus takes the central position.

Thus 98765.432 means
9.104 + 8.103 +7.102 + 6.101 + 5.10° + 4.10-1 + 3.10-2 +2.10-3.

255. In denoting any number in this way, a cypher, which has no figures on one side of it except cyphers, is called Insignificant. The other figures are called Significant.

256. Since our system of notation has ten for its base, it is tremely convenient to use ten for the base of our logarithms. Thus, writing the minus sign over the negative logarithms,

10°= 1,

.. logio 1 = 0, 101 = 10, .. logi 10 = 1,


.. log10 0.1 =ī, 102 = 100,

.. logio 100 = 2, 10-= 0.01, i logo 0.01 = 2, 103 = 1000, .. logio 1000 = 3, 10-3 = 0·001, .. logio 0.001 = 3,

and so on. Any integral power of 10 is expressed by a unit and a number of zeros.

Its logarithm is equal to the number of removes of the unit from the units' place. [Or if, as above, we fill in the units' place with a zero, the logarithm of any integral power of ten is equal to the number of zeros by which that power is expressed.] The logarithm is positive or negative, according as the removal of the unit is backwards or forwards. J. T.


257. The logarithm of any number, which is not an integral power of 10, will be fractional, and, if expressed in decimal notation, will contain an integral and a decimal part.

The method of calculating the value of logarithms will be explained in a later chapter.

The logarithms to the base 10 of all rational numbers, except those expressed by unity and zeros, are irrational.

Thus the logarithm of 3 to the base 10 is '4771213 correctly to 7 places of decimals.

This means that 10.4771213=3 approximately, i.e. that

104771213=310000000 approximately. The student is not recommended to test the truth of this statement by multiplying 3 by itself 10000000 times.

Negative Logarithms. 258. Since log10 1 = 0, the logarithms of all numbers less than 1 will be less than 0, i.e. negative. See Art. 243, Cor.

A negative logarithm, containing an integral and decimal part, is conveniently expressed in a form in which the integral part alone is negative.

- (3•12564)=-3 – •12564 = - 4+1 - •12564=-4+ •87436.

This is written 4.87436 : the minus sign being written over the integer in order to indicate that the integer alone is negative.

259. DEF. 1. The decimal part, expressed positively, of a logarithm is called its Mantissa.

DEF. 2. The integral part, found after expressing the mantissa positively, of a logarithm is called its Characteristic.

Thus the logarithm of .00074879 is - 3.12564. But the mantissa of this logarithm is not – •12564, nor is its characteristic – 3.

We first express it in the form 4.87436 in which the mantissa is positive. Thus the required mantissa is •87436 and the characteristic is – 4.

260. The special convenience of using 10 as the base of Logarithms will be shown when we have proved two rules relating respectively to the Characteristic and Mantissa.

The Rule of the Characteristic. 261. The characteristic of the logarithm to the base 10 of any number expressed in decimal notation, may be seen by inspection from the position of the first significant figure with relation to the units' place in the given number. Thus:

The characteristic is equal to the number of removes of the first significant figure from the units place:--being positive or negative according as the removal is backwards or forwards.

For, let s be the first significant figure: and n the number of its removes from the units' place.

Then the position of s gives it the value s.10”.
Hence the given number is > 10n and < 109+1.
.. its logarithm to base 10 > n and <n+1.

For, by Art. 243, Cor., since 10 > 1, the logarithms of numbers to the base 10 increase as the numbers increase.

But the mantissa is always a positive quantity < 1.
Hence the characteristic of the logarithm is n.

Examples. The characteristic of the logarithm of 32572 is +4 (the first significant figure 3 being four places to the left of the units' place).

For 32572 > 10000 and < 100000 ; i.e. > 104 but < 105.

The characteristic of the logarithm of .00032572 is – 4 (the figure 3 being four places to the right of the units' place).

For •00032572 > .0001 and < .001; i.e. > 10-4 but < 10-3.

The characteristic of the logarithm of 325•72 is + 2 (the figure 3 being two places to the left of the units' place).

For 325•72 > 100 but 1000, i.e. > 102 but < 103.

The Rule of the Mantissa. 262. The mantissue of the logarithms of two numbers are the same, if the numbers, expressed in decimal notation, have the same series of significant figures.

For such numbers differ only in the position of the decimal point. The larger is, therefore, obtained from the smaller by multiplying it by some positive integral power of 10.

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